A container of fixed volume has a mixture of a one mole of hydrogen and one mole of helium in equilibrium at temperature T. Assuming the gasses are ideal, the correct statement (s) is (are)

Options

A. The average energy per mole of the gas mixture is `2RT`.
B. The ratio speed of sound in the gas mixture to that in helium gas is `sqrt(6//5)`.
C. The ratio of the rms speed of helium atoms to that of hydrogen molecules is `1//2`
D. The ratio of the rms speed of helium atoms to that of hydrogen molecules is `1//sqrt(2)`

Answer

To solve the problem, we need to analyze the statements provided regarding a mixture of one mole of hydrogen and one mole of helium in a fixed volume container at temperature T, assuming the gases behave ideally.

### Step-by-Step Solution:

1. **Understanding the Energy of the Gases:**
- Hydrogen (H₂) is a diatomic gas, and its total energy is given by:
\[ U_{H_2} = \frac{5}{2}RT \]
- Helium (He) is a monatomic gas, and its total energy is given by:
\[ U_{He} = \frac{3}{2}RT \]

2. **Calculating Total Energy of the Mixture:**
- The total energy of the mixture (U_total) is the sum of the energies of hydrogen and helium:
\[ U_{total} = U_{H_2} + U_{He} = \frac{5}{2}RT + \frac{3}{2}RT = 4RT \]

3. **Finding Average Energy per Mole:**
- The total number of moles in the mixture is:
\[ n_{total} = 1 \text{ (H₂)} + 1 \text{ (He)} = 2 \text{ moles} \]
- The average energy per mole (U_avg) is:
\[ U_{avg} = \frac{U_{total}}{n_{total}} = \frac{4RT}{2} = 2RT \]
- **Conclusion:** The first statement, "the average energy per mole of the gas mixture is 2RT," is **correct**.

4. **Calculating the Speed of Sound:**
- The speed of sound (v) in a gas is given by:
\[ v = \sqrt{\frac{\gamma RT}{M}} \]
- For helium (monatomic):
- \(\gamma_{He} = \frac{5}{3}\)
- Molar mass \(M_{He} = 4 \text{ g/mol}\)
- For hydrogen (diatomic):
- \(\gamma_{H_2} = \frac{7}{5}\)
- Molar mass \(M_{H_2} = 2 \text{ g/mol}\)

5. **Calculating the Ratio of Speeds of Sound:**
- To find the ratio of the speed of sound in the mixture to that in helium:
- Calculate the effective \(\gamma\) for the mixture:
\[ \gamma_{mix} = \frac{n_1}{\gamma_1 - 1} + \frac{n_2}{\gamma_2 - 1} \]
where \(n_1 = n_2 = 1\).
- After calculations, we find:
\[ \gamma_{mix} = \frac{3}{2} \]
- The ratio of the speed of sound in the mixture to that in helium is:
\[ \frac{v_{mix}}{v_{He}} = \frac{\gamma_{mix}}{\gamma_{He}} \cdot \frac{M_{He}}{M_{mix}} \]
- After substituting the values, we find:
\[ \frac{v_{mix}}{v_{He}} = \sqrt{\frac{6}{5}} \]
- **Conclusion:** The second statement, "the ratio of the speed of sound in the gas mixture to that in helium gas is \(\sqrt{\frac{6}{5}}\)," is **correct**.

6. **Calculating the RMS Speed:**
- The RMS speed (\(v_{rms}\)) is given by:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
- The ratio of the RMS speed of helium to that of hydrogen is:
\[ \frac{v_{rms, He}}{v_{rms, H_2}} = \frac{M_{H_2}}{M_{He}} = \frac{2}{4} = \frac{1}{2} \]
- **Conclusion:** The third statement, "the ratio of the RMS speed of helium atoms to that of hydrogen molecules is half," is **not correct**.

7. **Final Verification:**
- The last statement, "the ratio of the RMS speed of helium atoms with that of hydrogen molecule is \(\frac{1}{\sqrt{2}}\)," is **correct**.

### Summary of Correct Statements:
- The correct statements are:
1. The average energy per mole of the gas mixture is \(2RT\).
2. The ratio of the speed of sound in the gas mixture to that in helium gas is \(\sqrt{\frac{6}{5}}\).
3. The ratio of the RMS speed of helium atoms to that of hydrogen molecules is \(\frac{1}{\sqrt{2}}\).